Discrete Structures

1. Discrete Structures Chapter 2: The Logic of Compound Statements 2.1 Logical Forms and Equivalence Logic is a science of the necessary laws of thought, without which no employment of the understanding and the reason takes place. – Immanuel Kant, 1724 – 1804 Foundation for the Metaphysics of Morals, 1785 2.1 Logical Forms and Equivalences

2. Logic • Logic is the study of reasoning; specifically whether reasoning is correct.Note: We use p, q, and r to represent propositions. • Logic does: • Assess if an argument is valid/invalid • Logic does not directly: • Assess the truth of atomic statements 2.1 Logical Forms and Equivalences

3. Statements • Statement or Proposition – any sentence that is true or false but not both 2.1 Logical Forms and Equivalences

4. Statements • Examples of Statements: • George Washington was the first president of the United States. • Baltimore is the capital of Maryland. • Seventeen is an even number. • The above statements are either true (1) or false (2, 3) 2.1 Logical Forms and Equivalences

5. Not Statements • These are not statements: • Earth is the only planet in the universe that contains life. • Buy two tickets to the rock concert for Friday • Why should we study logic? • None of the above can be determined to be true or false so they are not statements. 2.1 Logical Forms and Equivalences

6. Ambiguity • It is possible that a sentence is a statement yet we can not determine its truth or falsity because of an ambiguity or lack of qualification. • Examples: • Yesterday it was cold. • He thinks Philadelphia is a wonderful city. • Lucille is a brunette. For (1) we need to determine what we mean by the word “cold” For (2) we need to know whose opinion is being considered. For (3) it depends on which Lucille we are discussing. 2.1 Logical Forms and Equivalences

7. Compound Statements • In speech and writing, we combine propositions using connectives such as and or even or. • For example, “It is snowing” and “It is cold” can be combined into a single proposition “it is snowing and it is cold.” 2.1 Logical Forms and Equivalences

8. Compound Statements • Let: p = “It is snowing.” q = “It is cold.” 2.1 Logical Forms and Equivalences

9. Translating: English to Symbols • The word but translates the same as and. “Jim is tall but he is not heavy” translates to “Jim is tall AND he is not heavy” 2.1 Logical Forms and Equivalences

10. Translating: English to Symbols • The words neither-nor translates the same as not. “Neither a borrower nor a lender be” translates to “Do NOT be a borrower and do NOT be a lender” 2.1 Logical Forms and Equivalences

11. Truth Values • If sentences are statements then the must have well-defined truth values meaning the sentences must be either true or false. 2.1 Logical Forms and Equivalences

12. Negation • Definition If p is a statement variable, the negation of p is “not p” or “it is not the case that p” and is denoted p. It has the opposite truth value from p: if p is true, p is false if p is false, p is true 2.1 Logical Forms and Equivalences

13. Negation Truth Table The truth value for negation are summarized in the truth table on the right. 2.1 Logical Forms and Equivalences

14. Conjunction • Definition If p and q are statement variables, the conjunction of p and q is “p and q”, denoted p  q. It is true when BOTH p and q are true. If either p or q is false, or both are false, p  q is false. 2.1 Logical Forms and Equivalences

15. Conjunction Truth Table • The truth value for conjunction are summarized in the truth table on the right. 2.1 Logical Forms and Equivalences

16. Disjunction (Inclusive Or) • Definition • If p and q are statement variables, the disjunction of p and q is “p or q”, denoted p  q. It is true when either p is true, q is true, or both p and q are true. If both p and q is false, p  q is false. • Example • You may have cream or sugar with your coffee 2.1 Logical Forms and Equivalences

17. Disjunction Truth Table • The truth value for conjunction are summarized in the truth table on the right. 2.1 Logical Forms and Equivalences

18. Exclusive Or • Definition • If p and q are statement variables, the exclusive or of p and q is “p or q”, denoted pq. It is true when either p is true or when q is true, but not both. If both p and q is false, p  q is false. If both p and q is true, p  q is false. • Example • Your meal comes with soup or salad. 2.1 Logical Forms and Equivalences

19. Exclusive Or Truth Table • The truth value for conjunction are summarized in the truth table on the right. 2.1 Logical Forms and Equivalences

20. Statement Form • Definition • A statement form is an expression made up of statement variables such as p, q, and r and logical connectives that becomes a statement when actual statements are substituted for the component statement variables. 2.1 Logical Forms and Equivalences

21. Example • Construct a truth table for the statement form (pq)(pq). 2.1 Logical Forms and Equivalences

22. Tautology • Definition • A tautology is a statement form that is always true regardless of the truth values of the individual statements substituted for its statement variables. 2.1 Logical Forms and Equivalences

23. Contradiction • Definition • A contradiction is a statement form that is always false regardless of the truth values of the individual statements substituted for its statement variables. 2.1 Logical Forms and Equivalences

24. Logically Equivalent • Definitions • Two statement forms are logically equivalentiff they have identical truth values for each possible substitution of statements for their statement variables. Logical equivalence of statement forms P and Q is denoted by PQ. 2.1 Logical Forms and Equivalences

25. Testing for Logical Equivalence • Construct a truth table with one column for the truth values of P and another column for the truth values of Q. • Check the combination of truth values of the statement variables. • If in each row the truth value of P is the same as the truth value of Q, then PQ. • If in each row the truth value of P is not the same as the truth value of Q, then PQ. 2.1 Logical Forms and Equivalences

26. Example Are the statement forms (pq) and pq logically equivalent?  (pq)  pq 2.1 Logical Forms and Equivalences

27. Theorem 2.1.1 – Logical Equivalence • Given any statement variables p, q, and r, a tautology t, and a contradiction c, the following logical equivalences hold. 2.1 Logical Forms and Equivalences

28. De Morgan’s Laws • The negation of an and statement is logically equivalent to the or statement in which each component is negated. (p  q)  p  q • The negation of the or statement is logically equivalent to the and statement in which each component is negated. (p  q)  p  q 2.1 Logical Forms and Equivalences

29. Example – pg. 38 # 33 • Use De Morgan’s Laws to write negations for the statement. -10 < x < 2 2.1 Logical Forms and Equivalences

30. Example – pg. 38 # 51 • Use Theorem 2.1.1 to verify the logical equivalences. Supply a reason for each step. p (q  p)  p 2.1 Logical Forms and Equivalences

31. Next time • Continue with logical equivalence. • Discuss valid and invalid arguments. 2.1 Logical Forms and Equivalences